Trigonometric functions of an acute angle

Trigonometric functions of acute angle: 
sine, cosine, tangent, cotangent, secant, cosecant. 
Exact values of trigonometric functions
for some most used acute angles.

Trigonometric functions of an acute angle are ratios of different pairs of sides of a right-angled triangle ( Fig.2 ).

   1)   Sine:  sin A = a / c   ( a ratio of an opposite leg o a hypotenuse ) .  

   2)   Cosine:  cos A = b / c  ( a ratio of an adjacent leg to a hypotenuse ) .

   3)   Tangent:  tan A = a / b   ( a ratio of an opposite leg to an adjacent leg ) .

   4)   Cotangent:  cot A = b / a   ( a ratio of an adjacent leg to an opposite leg ) .

   5)   Secant:   sec A = c / b   ( a ratio of a hypotenuse to an adjacent leg ) .

   6)  Cosecant:  cosec A = c / a   ( a ratio of a hypotenuse to an opposite leg ) .

There are analogous formulas for another acute angle B ( Write them, please ! ).

E x a m p l e .  A right-angled triangle ABC  ( Fig.2 ) has the following legs:
                       a = 4,  b = 3. Find sine, cosine and tangent of angle A.

S o l u t i o n .  At first we find a hypotenuse, using Pythagorean theorem:

                       c ² = a ² + b ², 

                       According to the above mentioned formulas we have:
                       sin A = a / c = 4 / 5;  cos A = b / c = 3 / 5;  tan A = a / b = 4 / 3. 

For some angles it is possible to write exact values of their trigonometric functions. The most important cases are presented in the table:

Although angles 0° and 90° cannot be acute in a right-angled triangle, but at enlargement of notion of trigonometric functions ( see below), also these angles are considered. A symbol  ∞  in the table means that absolute value of the function increases unboundedly, if the angle approaches the shown value.

مواضيع ذات صلة

Trigonometric inequalities

Trigonometric functions of any angle

Trigonometric equations. Main methods for solving

Transforming of trigonometric expressions to product

Transforming of degree measure to radian one and back

Systems of simultaneous trigonometric equations

Some important correlations

Solving of right-angled triangles

Relations between trigonometric functions of the same angle

Reduction formulas

Radian and degree measures of angles

Problems: Trigonometric transformations